The finite element approach has been previously used, with the help of the ATILA code, to model the scattering of acoustic waves by single or doubly periodic passive structures ͓A. C. Hladky-Hennion et al., J. Acoust. Soc. Am. 90, 3356 -3367 ͑1991͔͒. This paper presents a new extension of this technique to the analysis of the propagation of plane acoustic waves in passive periodic materials without losses and describes with particular emphasis its application to doubly periodic materials containing different types of inclusions. In the proposed approach, only the unit cell of the periodic material has to be meshed, thanks to Bloch-Floquet relations. The modeling of these materials provides dispersion curves from which results of physical interest can be easily extracted: identification of propagation modes, cutoff frequencies, passbands, stopbands, as well as effective homogeneous properties. In this paper, the general method is first described, and particularly the aspects related to the periodicity. Then a test example is given for which analytical results exist. This example is followed by detailed presentations of finite element results, in the case of periodic materials containing inclusions or cylindrical pores. The homogenized properties of porous materials are determined with the help of an anisotropic model, in the large wavelength limit. A validation has been carried out with periodically perforated plates, the resonance frequencies of which have been measured. The efficiency and the versatility of the method is thus clearly demonstrated.
The finite element approach has been previously used, with the help of the ATILA code, to model the scattering of acoustic waves by single periodic structures, such as compliant tube gratings [A. C. Hennion et al., J. Acoust. Soc. Am. 87, 1861-1870 (1990) ]. In this paper, the same approach is extended to doubly periodic structures, such as Alberich anechoic coatings. To do this, only the unit cell of the periodic structure, including a small part of the surrounding fluid domain, has to be meshed, due to the use of classical Bloch type relations. Then, the effects of the remaining fluid domain are accounted for by matching the pressure field in the finite element mesh with simple plane-wave expansions of the ingoing and outgoing waves. After an outline of the method, the paper describes the results obtained for the scattering of a plane wave by different periodic structures. Internal losses are taken into account and the incident plane wave impinges at normal or oblique incidence. Numerical results obtained for Alberich anechoic coatings are first analyzed to check the convergence and then compared to previous numerical results or to experimental results, demonstrating that the finite element approach is accurate and well suited to predict the behavior of these gratings. Moreover, careful attention is devoted to the analysis of the inclusion vibrations, to identify the origin of the resonance mechanisms. PACS numbers: 43.20.Fn, 43.20.Bi, 43.30.Gv, 43.30.Ky INTRODUCTION The scattering of a plane acoustic wave by a periodic array of elastic structures is widely used in underwater acoustics. In fact, such arrays can be efficient as reflecting screens or absorbers, within a large frequency band, and are used, for example, to increase the directivity and the acoustic level of low-frequency sources, to insulate receiving hydrophones from noise sources which are in proximity or to provide anechoic properties to the walls of acoustic tanks. The immersed periodic structures can be split into two groups. The first one contains single periodic structures, such as single or double layered compliant tube gratings, which are directly immersed or embedded in a viscoelastic medium. In this case, and in the frequency band of interest, the incident plane wave excites a resonance mode of the tubes and the grating behaves as a reflecting baffle. •-9 The second group contains doubly periodic structures such as Alberich anechoic coatings. •ø-•4 These coatings are multilayered structures in which one or several layers made up of absorbing materials contain doubly periodic inclusions, such as spherical or cylindrical cavities. Then, in the frequency band of interest, the incident wave excites a resonance mode of the inclusions and the coating behaves as a sound absorber. In order to explain the physical behavior and to help the design of such structures, several authors have built accurate mathematical models, which provide insertion loss values in nice agreement with measurements. On the one Associated with the CNRS, U.R.A. 253. hand,...
A two-dimensional mathematical model has been developed to analyze the scattering of plane acoustic waves from an infinite, uniform, plane grating of compliant tubes. It relies upon the finite element method and uses the ATILA code [J. N. Decarpigny et al., J. Acoust. Soc. Am. 78, 1499 (1985) ]. To do this, only the unit cell of the periodic structure, including a small part of the surrounding fluid domain, has to be meshed, thanks to the Bloch-Floquet theorem, and the effects of the remaining fluid domain are accounted for by matching the pressure field in the fi'nite element mesh with simple plane wave expansions of the ingoing and outgoing waves. This paper describes results obtained for the scattering of a plane wave from different tube gratings, including internal losses, at oblique incidence. Comparing finite element results to analytical or experimental results allows for the validation of the model. Then, various compliant tube gratings are considered to demonstrate the efficiency and versatility of this approach. Finally, the generalization to doubly periodic gratings is emphasized. PACS numbers' 43.20.Fn, 43.20.Bi, 43.30.Gv
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